Question

The family of integral curves of the differential equation $\frac{d y}{d x}+x^{3} y=x$ is cut by the line $x=2 ;$ the tangents at the points of intersection are concurrent at $(\lambda, \mu)$. Then find the value of $\left[\frac{\lambda}{\mu}\right]$, where $[\,\,.\,\,]$ denotes greatest integer function.

Answer 1

$\frac{d y}{d x}+x^{3} y=x$
Equation of tangent at $(2, \alpha)$
$(y-\alpha)=(2-8 \alpha)(x-2)$
$\Rightarrow y-\alpha=2 x-4-8 \alpha(x-2)$
$\Rightarrow(y-2 x+4)+\alpha(8 x-17)=0$
$x=\frac{17}{8} \text { and } y=2 x-4$
$\Rightarrow y=\frac{1}{4}$
$\therefore (\lambda, \mu) \equiv\left(\frac{17}{8}, \frac{1}{4}\right)$
$\therefore \frac{\lambda}{\mu}=\frac{17}{2}$
$\Rightarrow\left[\frac{\lambda}{\mu}\right]=8$